You want a standard error for the mean of 2,000 daily returns that carry a lag-1 autocorrelation of 0.3. Say what the iid bootstrap gets wrong here, quantify the error, and choose a block length for a moving block bootstrap with the reasoning.

You want a standard error for the mean of 2,000 daily returns that carry a lag-1 autocorrelation of 0.3. Say what the iid bootstrap gets wrong here, quantify the error, and choose a block length for a moving block bootstrap with the reasoning.

Approach: Compare the variance of a sample mean under an AR(1) with the iid formula, then pick a block length from the standard rate for the moving block bootstrap.

The true standard error is about 36% larger than the iid bootstrap reports, and a moving block bootstrap with blocks of about 13 days fixes it. For an AR(1) with lag-one autocorrelation rho the long-run variance factor is (1 + rho)/(1 - rho) = 1.3/0.7 = 1.857, so the true variance of the sample mean is 1.857 times sigma^2/n and the true standard error is sqrt(1.857) = 1.36 times the iid one. Resampling single days independently destroys the serial dependence, so the bootstrap distribution reproduces the iid figure and every interval built from it is too narrow at exactly that ratio. The moving block bootstrap resamples contiguous blocks of length l, keeping dependence inside a block and losing only the dependence across joins, and it is consistent provided l grows with n more slowly; the standard rate l proportional to n^{1/3} gives 2000^{1/3} = 12.6. That is about 13 days. Blocks that are too short leave the same understatement, and blocks that are too long leave few distinct blocks and a very noisy bootstrap variance.

Follow-up: Under a GARCH process with no autocorrelation in returns at all, does the iid bootstrap give the right standard error for the mean, and for the variance?

Key concepts: block bootstrap, serial dependence, long-run variance, autocorrelation.